Irregular polygons - Equal area under horizontal line

In summary, irregular polygons are geometric shapes with unequal angles and sides. The equal area under a horizontal line in irregular polygons is important for consistent measurement of surface area, which is useful in land surveying, physics, and engineering. This area is calculated by dividing the shape into smaller, regular polygons and summing their areas. Irregular polygons can have equal area under a horizontal line but different perimeters, and this concept has various real-world applications in fields such as architecture and construction.
  • #1
jamnitzer
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I've been trying to find an algorithm for finding a line passing through any 2-dimensional polygon that will divide the shape horizontally or vertically into two sections that each have the exact same area, or rather exactly half the area of the whole. I've prepared some simple examples below.

My question is if someone has heard of this or done something similar, and/or if anyone knows where to find more information.

Using the shoelace method of measuring area and/or after calculating the center of mass, I want to find a way to find either a horizontal line that will divide the top and bottom of a shape or a vertical line that will divide left and right such that the area of the shape on either side will be equal.

So the half-way line in terms of area, but only from one direction. I'm hoping there's already a name for this somewhere and I just haven't been able to find it. If this is more of a calculus question let me know. Really any information will be appreciated.


Some simple examples:

These are two shapes defined with their coordinates (x,y), repeating the first at the end.
{(4,0),(6,0),(6,2),(8,4),(4,4),(4,6),(2,4),(0,4),(4,0)} Area = 20
{(1,0),(1,1),(7,1),(4,0),(8,0),(8,1),(6,4),(4,2),(2,4),(6,6),(8,8),(0,8),(0,4),(1,0)} Area = 40.5

For the first shape, the vertical line evenly dividing left and right the area is x= 4.
The horizontal line dividing top and bottom is y= 4/sqrt(2).

In the second shape, the vertical line is x= -(sqrt(109)-20)/3
The horizontal line is y= (sqrt(2)sqrt(5.5)+6)/2
 
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  • #2


Thank you for your interesting question about finding a line that divides a 2-dimensional polygon into two sections with equal area. This type of problem falls under the field of computational geometry and there are several algorithms that can help you achieve this goal.

One approach is to use the centroid of the polygon to find the line of symmetry. The centroid is the point at which the polygon would balance if it were cut out of a sheet of uniform density material. To find the centroid, you can use the shoelace method or any other method for finding the center of mass.

Once you have the centroid, you can draw a line passing through it that divides the polygon into two sections with equal area. This line will be perpendicular to the line connecting the centroid to any vertex of the polygon.

Another approach is to use a binary search algorithm. This involves drawing a line through the polygon and calculating the area of each section on either side of the line. If the areas are unequal, you can adjust the position of the line and recalculate until you find a line that divides the polygon into two sections with equal area.

There are also more complex algorithms, such as the sweep line algorithm, that can be used to find a line of symmetry for a 2-dimensional polygon.

I recommend researching these algorithms and determining which one best suits your needs. You may also find helpful resources by searching for terms such as "symmetry line algorithm" or "equal area division of polygon."

I hope this information helps in your search for a solution. Good luck!
 

FAQ: Irregular polygons - Equal area under horizontal line

What are irregular polygons?

Irregular polygons are closed geometric shapes with three or more sides that do not have equal angles or equal length sides. They can be convex or concave and have varying degrees of symmetry.

What is the importance of equal area under horizontal line in irregular polygons?

The equal area under a horizontal line in irregular polygons is important because it allows for a consistent measurement of the shape's surface area. This is particularly useful in applications such as land surveying or calculating the surface area of irregularly shaped objects in physics or engineering.

How is the equal area under horizontal line calculated in irregular polygons?

The equal area under horizontal line in irregular polygons is calculated by dividing the shape into smaller, regular polygons and calculating the area of each smaller polygon. These areas are then summed together to get the total area under the horizontal line.

Can irregular polygons have equal area under horizontal line but different perimeters?

Yes, irregular polygons can have equal area under a horizontal line but different perimeters. This is because the equal area under the horizontal line only takes into account the surface area of the shape, while the perimeter measures the length of the shape's boundary.

How can equal area under horizontal line be useful in real-world applications?

The concept of equal area under horizontal line in irregular polygons has many practical applications. It can be used in land surveying, architecture and construction, and even in calculating the surface area of irregularly shaped objects in fields like physics and engineering.

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